INVERSE METHOD OF DEFINITE INTEGRALS. 143 



{p, 9),„ = H,r - f . ^H,,f" + ^^^ >\^^H,.r 



on the supposition that we put h = after the operations above indicated, 

 are performed. 



Separate in this expression the symbols of operation and of quantity, 

 and we shall obtain the equation 



(p,q),„ = (l-fr.H,J'': 



But \U — 1 or \^ T- x//° indicates that we must subtract the original 

 value of Hp,q, from the value it receives when h + 1 is put for h, 

 that is, it is the same as performing the operation A of finite differences ; 

 this consideration transforms the preceding equation, to this 



(p, q)m = (-iy" A" . Hf.qt"", when h is put =0. 



In like manner if we put 



„ ^ i,qh-\-\) {qh + l+p) {qh + 1 -\-{m-l) .p} 



"■' 1(1+^) {\ + {m-\).p} 



we have (g-, jo)„ = (-l)"' A'" Jf^.pi?"', when A = 0. 



Now observing that by the nature of reciprocal functions we have 



S* ip, q)m (q, p)n = 0, except when m = n, 



and by Art. 13., fi{p, q\{q,p)^ 



_ ip, q)'" 1.1.2.2.3. 3 .m . m 



~ 1 + mip+q) '1.1. (l+ju)(l + 9)(l + 2^)(l+29)...{l + (»w-l) .p} {l + {m-l).q} ' 



then since f{t) = «„ (p, q)o + «i (p, q)i + a, {p, q)^ + &c. 



we have ftf{t) . (q, p)„ 



_ (pq)'" 1.1.2.2. 3 m . m 



"**"'• l + m(p+q)'l.l{l+p){l+q) {l + im-l).p} {l+{m-l).q} ' 



t2 



